## [An "application" of Existential Closedness / the Nullstellensatz]{#An_"application"*of_Existential_Closedness*/_the_Nullstellensatz .mw-headline}[]

When originally writing the article, I tried to include the following fact as a corollary of the existential closedness of algebraically closed fields in the class of fields:

If *V* is a geometrically integral variety over a field *K* (i.e., the base change of *V* to the algebraic closure of *K* is integral), then every base change of *V* is integral.

The proof ended up getting a bit out of hand. I'm moving it to this talk page since it got a bit off topic for this article:

As another "application", let us prove:

**Corollary:** Let *K* be an algebraically closed field, and *L* be a field extending *K*. Let *A* be a *K*-algebra, possibly infinite dimensional. Suppose *A* is integral. Then so is *A*⊗_{K}*L*.

(In algebro-geometric terms, this means that if a variety is geometrically integral, it remains integral under any base change.)

*Proof:* We will instead prove a slightly stronger statement: let *V* and *W* be *K*-vector space, and let *μ* : *V*⊗_{K}*V* → *W* be some *K*-bilinear pairing. Tensoring everything with *L*, we get

*μ*_{L} : (*V*⊗_{K}*L*)⊗_{L}(*V*⊗_{K}*L*) → *W*⊗_{K}*L*

Suppose that *μ*(*v*_{1}, *v*_{2}) ≠ 0 whenever *v*_{1} ≠ 0 ≠ *v*_{2}. Then *μ*_{L} has the same property.

To see this, write V as a directed limit of finite dimensional submodules

$V = \bigcup\limits_{i \in I}V_{i}$

Let *W*_{i} ⊂ *W* be the *K*-span of *μ*(*V*_{i}, *V*_{i}), so that *μ* induces a map *V*_{i} ⊗ *V*_{i} → *W*_{i} for each *i* ∈ *I*, and the two maps

*V*_{i} ⊗ *V*_{i} → *W*_{i} ↪ *W*

*V*_{i} ⊗ *V*_{i} ↪ *V* ⊗ *V* → *W*

agree.

The functor − ⊗_{K}*L* is a left adjoint, so it preserves direct limits. Also, because *K* is a field, *L* is flat, so this functor preserves injections. Therefore,

$V \otimes_{K}L = \bigcup\limits_{i \in I}(V_{i} \otimes_{K}L)$

So, if there exist two non-zero vectors *v*_{1}, *v*_{2} ∈ *V*⊗_{K}*L* with *μ*_{L}(*v*_{1}, *v*_{2}) = 0, then some *V*_{i}⊗_{K}*L* already contains *v*_{1}, *v*_{2}.

Replacing *V* with *V*_{i} and *W* with *W*_{i}, we may assume that *V* and *W* are finite dimensional. In fact, we may assume that they are *K*^{n} and *K*^{m} for some *n* and *m*. Then *μ* is described by structure coefficients:

$\mu(e_{i},e_{j}) = \sum\limits_{k}c_{ijk}e_{k}$

where the *e*_{i} are standard basis vectors. The same structure coefficients are correct after tensoring with *L*.

Our assumption means that in *K*, the following system of equations cannot be solved:

$\bigwedge\limits_{k = 1}^{m}\sum\limits_{i,j}c_{ijk}v_{1}^{i}v_{2}^{j}$

$\prod\limits_{i = 1}^{n}v_{1}^{i} \neq 0$

$\prod\limits_{i = 1}^{n}v_{2}^{i} \neq 0$

in the variables *v*_{1}^{1}, *v*_{1}^{2}, …, *v*_{1}^{n}, *v*_{2}^{1}, *v*_{2}^{2}, …, *v*_{2}^{n}. By existential closedness of *K* in *L*, the same system of equations cannot be solved in *L*, which is what we wanted to show. QED.

Will Johnson (talk) 07:15, November 15, 2013 (UTC) ::: ::: :::